Explanation Given information: The mass of the package ( m ) is 50 kg . The velocity of the package ( v 1 ) is 2 m / s . The distance between the package and cable ( l ) is 8 m . The angle of spring ( θ ) is 20 ° . The spring constant ( k ) is 30 kN / m . The coefficient of kinetic friction between the package ( μ k ) is 0.15. The compression of the spring ( e 1 ) is 50 mm or 0 .05m . The acceleration due to gravity ( g ) is 9.81 m / s 2 . Calculation: Consider the position 1. Assume the position 1 be the initial position which 8 m from the end of spring when it is compressed of 50 mm by cable. Calculate the kinetic energy ( T 1 ) in the position 1 using the formula: T 1 = 1 2 m v 1 2 Substitute 50 kg for m and 2 m / s for v . T 1 = 1 2 × 50 × ( 2 ) 2 = 100 J Calculate the height ( h 1 ) from base to package using the relation: h 1 = l sin θ Substitute 8 m for l and 20 ° for θ . h 1 = ( 8 ) sin 20 ° = 2.7362 m Calculate the potential energy stored due to elongation at position 1 ( V 1 ) e using the formula: ( V 1 ) e = 1 2 k e 1 2 Substitute 0.050 m for e 1 and 30 kN / m for k . ( V 1 ) e = 1 2 × 30 × 10 3 × ( 0.050 ) 2 = 37.5 J Calculate the potential energy stored due to gravitation at position 1 ( V 1 ) g using the formula: ( V 1 ) g = m g h 1 Substitute 50 kg for m , 9.81 m / s 2 for g , and 2.7362 m for h 1 . ( V 1 ) g = 50 × 9.81 × ( 2.7362 m ) = 1342.1 J Calculate the potential energy stored in rod in position 1 ( V 1 ) using the relation: V 1 = ( V 1 ) e + ( V 1 ) g Substitute 37.5 J for ( V 1 ) e and 1342.1 J for ( V 1 ) g . V 1 = 37.5 J+1342 .1J = 1379.6 J Consider the position 2. Assume the position 2 be the maximum compression position. Calculate the kinetic energy ( T 2 ) in the position 2 using the formula: T 2 = 1 2 m v 2 2 Substitute 50 kg for m and 0 for v . T 2 = 1 2 × 50 × ( 0 ) 2 = 0 Calculate the height ( h 2 ) using the relation: h 2 = − x sin θ Here, x is the addition maximum deformation of the spring. Substitute 20 ° for θ . h 2 = − x sin 20 ° = − 0.34202 x Calculate the compression of the spring e 2 in position 2 using the relation: e 2 = e 1 + x Substitute 0.050 m for e 1 . e 2 = 0.050 + x Calculate the potential energy stored due to elongation at position 2 ( V 2 ) e using the formula: ( V 2 ) e = 1 2 k e 2 2 Substitute 0.050 + x for e 2 and 30 kN / m for k

Vector Mechanics for Engineers: Statics and Dynamics

12th Edition

ISBN: 9781259638091

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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Chapter 13.2, Problem 13.69P

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Find the maximum additional deformation of the spring (x) in bringing the package to rest.

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Chapter 13.2, Problem 13.68P

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A spring is used to stop a 50-kg package which is moving down a 20° incline. The spring has a constant k =34 kN/m and is held by cables so that it is initially compressed 50 mm. Knowing that the velocity of the package is 2 m/s when it is 8 m from the spring and neglecting friction, determine the maximum additional deformation of the spring in bringing the package to rest. Assume that the kinetic coefficient of friction between the package and the incline is 0.2. 2 m/s 50 kg 8m Cable 20° The maximum additional deformation of the spring in bringing the package to rest is 0.574 x m.

SHOW COMPLETE SOLUTION A spring is used to stop 50kg block moving down a 20 degrees inclined plane. The spring has a constant K= 30KN/M and is held by the cables so that it is initially compressed 50mm. Knowing that the velocity of the package is 2m/s when it is 8m from the spring and assuming the kinetic coefficient of friction between the package and the incline is 0.20. Determine: D. Final total potential energy E. Initial kinetic energy F. Initial total potential energy

Question 1. A spring is used to stop a 60 kg package which is sliding on a horizontal surface. The spring has a constant k = 20 kN/m and is held by cables so that it is initially compressed 120 mm. The package has a velocity of 2.5 m/s in the position shown and the maximum deflection of the spring is 40 mm. Determine (a) the coefficient of kinetic friction between the package and surface and (b) the velocity of the package as it passes again through the position shown. 2.5 m/s Cable 60 kg- 600 mm-

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Vector Mechanics for Engineers: Statics and Dynamics

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Find the maximum additional deformation of the spring ( x ) in bringing the package to rest.

Solution Summary: The author calculates the maximum additional deformation of the spring in bringing the package to rest using the formula: underset_0.2012m

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